Loss Modle

LOSS MODELS: FROM DATA TO DECISIONS, 3RD ED.

LOSS MODELS: FROM DATA TO DECISIONS, 3RD ED.

Stuart A. Klugman
Drake University

Harry H. Panjer
University of Waterloo

Gordon E. Willmot
University of Waterloo

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A JOHN WILEY & SONS, INC., PUBLICATION

CONTENTS

Preface PART I 1 Modeling 1.1 The model-based approach 1.1.1 The modeling process 1.1.2 Themodeling advantage Organization of this book INTRODUCTION

xvii

3 3 4 5 6 9 9 11 20 21 21 28 29 30 31
v

1.2 2

Random variables 2.1 2.2 Introduction Key functions and four models 2.2.1 Exercises

3

Basic distributional quantities 3.1 3.2 3.3 Moments 3.1.1 Exercises Quantiles 3.2.1 Exercises Generating functions and sums of random variables

vi

CONTENTS

3.4

3.5

3.3.1Exercises Tails of distributions 3.4.1 Classication based on moments 3.4.2 Comparison based on limiting tail behavior 3.4.3 Classication based on the hazard rate function 3.4.4 Classication based on the mean excess loss function 3.4.5 Equilibrium distributions and tail behavior 3.4.6 Exercises Measures of Risk 3.5.1 Introduction 3.5.2 Risk measures and coherence 3.5.3 Value-at-Risk 3.5.4Tail-Value-at-Risk 3.5.5 Exercises PART II ACTUARIAL MODELS

35 35 36 37 37 39 40 41 43 43 43 45 47 51

4

Characteristics of Actuarial Models 4.1 4.2 Introduction The role of parameters 4.2.1 Parametric and scale distributions 4.2.2 Parametric distribution families 4.2.3 Finite mixture distributions 4.2.4 Data-dependent distributions 4.2.5 Exercises

55 55 56 56 58 58 60 62 65 65 65 66 66 68 6872 73 75 78 78 79 79 81

5

Continuous models 5.1 5.2 Introduction Creating new distributions 5.2.1 Multiplication by a constant 5.2.2 Raising to a power 5.2.3 Exponentiation 5.2.4 Mixing 5.2.5 Frailty models 5.2.6 Splicing 5.2.7 Exercises Selected distributions and their relationships 5.3.1 Introduction 5.3.2 Two parametric families 5.3.3 Limiting distributions 5.3.4 Exercises

5.3CONTENTS

vii

5.4 5.5

5.6

The linear exponential family 5.4.1 Exercises TVaR for continuous distributions 5.5.1 Continuous elliptical distributions 5.5.2 TVaR for the linear exponential family 5.5.3 Exercise Extreme value distributions 5.6.1 Introduction 5.6.2 Distribution of the maximum 5.6.3 Stability of the maximum of the extreme value distribution 5.6.4 The Fisher——Tippett theorem 5.6.5Maximum domain of attraction 5.6.6 Generalized Pareto distributions 5.6.7 Stability of excesses of the generalized Pareto 5.6.8 Limiting distributions of excesses 5.6.9 TVaR for extreme value distributions 5.6.10 Further reading 5.6.11 Exercises

82 84 84 85 87 89 89 89 91 95 96 98 101 103 104 105 106 106 109 109 110 110 113 115 117 120 120 120 123 125 128 129 130 135 135 142 142 147 147 1476

Discrete distributions and processes 6.1 6.2 6.3 6.4 6.5 6.6 Introduction 6.1.1 Exercise The Poisson distribution The negative binomial distribution The binomial distribution The ( 0) class 6.5.1 Exercises Counting processes 6.6.1 Introduction and denitions 6.6.2 Poisson processes 6.6.3 Processes with contagion 6.6.4 Other processes 6.6.5 Exercises Truncation and modication at zero 6.7.1Exercises Compound frequency models 6.8.1 Exercises Further properties of the compound Poisson class 6.9.1 Exercises Mixed frequency distributions 6.10.1 General mixed frequency distribution

6.7 6.8 6.9 6.10

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CONTENTS

6.11 6.12 6.13 6.14

6.10.2 Mixed Poisson distributions 6.10.3 Exercises Mixed Poisson processes 6.11.1 Exercises E ect of exposure on frequency An inventory ofdiscrete distributions 6.13.1 Exercises TVaR for discrete distributions 6.14.1 TVaR for the discrete linear exponential family 6.14.2 Exercises

149 154 156 160 162 163 164 166 166 170 171 171 172 174 174 175 175 176 181 181 184 184 187 188 189 189 190 194 195 197 197 199 199 201 203 207

7

Multivariate models 7.1 7.2 7.3 Introduction Sklar’’s theorem and copulas Measures of dependency 7.3.1...